AD-A242 595

NASA Contractor Report 189046ICASE Report No. 91-78

DT!CNOV 1 8 1991- IICASE S NV19

ENERGY MODELS FOR ONE-CARRIER TRANSPORTIN SEMICONDUCTOR DEVICES

Joseph W. JeromeChi-Wang Shu

Contract No. NASI-18605October 1991

Institute for Computer Applications in Science and EngineeringNASA Langley Research CenterHampton, Virginia 23665-5225

Operated by the Universities Spdce Research Association

This document has been approved[ for P',blic ,,deaz '? nd s ,:iiq" 91-15746,IInnAlnii rf1, a i! II

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LRngley ReAeorch CenterHinptnn.Virginta ?1F)9 1 5225

91 111.5 066

ENERGY MODELS FOR ONE-CARRIERTRANSPORT IN SEMICONDUCTOR DEVICES

Accesion ForJoseph W. Jerome' RNTIS1 CRAI&I

Department of Mathematics D-FiC TA3Northwestern University U:aioti ced

Evanston, IL 60208By

Chi-Wang Shu2 Dist ib*Alo;i I

Division of Applied Mathematics Avai,,iliy Coct

Brown University Av:;I ,nu t or

Providence, RI 02912 Dist SPcia(

ABSTRACT __l

Moment models of carrier transport, derived from the Boltzmann equation, have made

possible the simulation of certain key effects through such realistic assumptions as energy

dependent mobility functions. This type of global dependence permits the observation ofvelocity overshoot in the vicinity of device junctions, not discerned via classical drift-diffusionmodels, which are primarily local in nature. It has been found that a critical role is playedin the hydrodynamic model by the heat conduction term. When ignored, the overshoot isinappropriately damped. When the standard choice of the Wiedemann-Franz law is made forthe conductivity, spurious overshoot is observed. Agreement with Monte-Carlo simulation inthis regime has required empirical modification of this law, as observed by IBM researchers,or nonstandard choices. In this paper, simulations of the hydrodynamic model in one andtwo dimensions, as well as simulations of a newly developed energy model, the RT model, willbe presented. The RT model, intermediate between the hydrodynamic and drift-diffusionmodel, was developed at the University of Illinois to eliminate the parabolic energy band andMaxwellian distribution assumptions, and to reduce the spurious overshoot with physicallyconsistent assumptions. The algorithms employed for both models are the essentially non-oscillatory shock capturing algorithms, developed at UCLA during the last dccade. Somemathematical results will be presented, and contrasted with the highly developed state of

the drift-diffusion model.1The first author is supported by the National Science Foundation under Grant DMS-8922398.2 This research was supported in part by the National Aeronautics and Space Administration under

NASA Contract No. NAS1-18605 while the second author was in residence at the Institute for ComputerApplications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665.Additional support was also provided by Army Research Office under Grant DAAL03-91-G-0123 and by theNational Aeronautics and Space Administration under Grant NAGI-1445.

I

1 Introduction

During the last decade, device modeling has attempted to incorporate general carrier heating,

velocity overshoot, and various small device features into carrier simulation. The popular

wisdom emerging from such concentrated study holds that global dependence of critical

quantities, such as mobilities, on energy and/or temperature, is essential if such phenomena

are to be modeled adequately. In this paper, we examine in detail the simulation of two

such energy models, including the hydrodynamic model and the RT model. We describe the

models, summarize some associated mathematical results, as well as the basic features of the

numerical algorithm, and then present the results of extensive numerical simulations for two-

dimensional MESFET devices, and for one-dimensional diodes. Both models represent one

carrier flow. The hydrodynamic model contains hyperbolic modes related to the momentum

equations, while the RT model does not possess such modes. In both cases, however, weemploy a conservation law format, and numerical methods suitable for such systems. The

ENO (essentially non-oscillatory) method employed makes use of adaptive stencils, and isparticularly adept at shock capturing if the parameter regime crosses from supersonic to

subsonic. Even if this does not occur, the convective terms are effectively discretized, via

this procedure, in both models. The first use of such methods in device simulation was in

[71, followed by the study (61, in which shocks were detected in micron devices at liquid Ni-

trogen temperatures, and at room temperature in shorter devices, by independent numerical

techniques.

Our development of the RT model follows that of [5]. These researchers attempted

to utilize a microscopic relaxation time approximation, which would allow for nonparabolic

energy bands and non-Maxwellian distribution functions. The approach allows for parameter

fitting of certain key quantities via Monte-Carlo simulation.

One of the principal conclusions of the paper is the essential dependence of the hydrody-

namic model upon the heat conduction term. Standard choices lead to numerically detected

spurious overshoot at the drain junction of an n- - n - n+ diode, while other choices signifi-

cantly damp this overshoot. Monte-Carlo simulations show that substantial underestimation

occurs when the heat conduction term is neglected. We refer the reader to [10], and to the

simulation results of this paper for amplification. The RT model was developed, partly in

response to the continuing debate concerning heat conduction processes in the hydrodynamic

model.

The status of mathematical results differs sharply between the hydrodynamic model,

on the one hand, and the drift-diffusion model on the other. For the former, we have

summarized two results, one by Gamba (cf. [81) for an idealized model, in which the adi-

1 m m lmnn u n la l lu llmmlml I II

abatic relation is employed, and another by Gardner, Jerome, and Rose (cf. [9]) in which

a Newton-Kantorovich theorem is developed for the n+ - n - n+ diode, yielding existence

and convergence in a specialized subsonic regime. The drift-diffusion model, on the other

hand, has been widely studied. Existence and approximation results have been carefully

developed, although uniqueness is still not well understood for this model. Existence for

the steady-state model is due in varying degrees of generality to many authors, including

Mock ([17]), Seidman ([21]), and the first author ([13]). A convergence theory, based upon

a calculus due to Krasnosel'skii, was presented in ([15]). Mathematical results have not yet

been developed for the strongly nonlinear RT model.

2 Hydrodynamic and Drift-Diffusion Models

2.1 Mass, momentum and energy transport equations

The equations as presented here are discussed in references [3], [20], and [4]. They are derived

as the first three moments of the Boltzmann equation, with the latter written for electrons

moving in an electric field asOf + -Vj - F -. j =C0-{+U.VYf -m .VUf=C. (1)

Here, f == f(, u, t) is the numerical distribution function of a carrier species, x is the

position vector, u is the species' group velocity vector, F = F(x, t) is the electric field, e

is the electron charge modulus, m is the effective electron mass, and C is the time rate of

change of f due to collisions. In the Boltzmann equation above, it has been assumed that the

traditional Lorentz force field does not have a component induced by an external magnetic

field. The moment equations, which will be derived subsequently, are expressed in terms of

certain dependent variables, where n is the electron concentration, v is the average velocity,

p is tile momentum density., P is the symmetric pressure tensor, q is the the heat flux, cj is

the internal energy, and C,, C,, and Cw represent moments of C, taken with respect to the

functions

ho(u) = 1

h11(u) mu,

The equat ioi is are gi '-!! by%:On + V. ( C) C, (2)

- ) + (p V)= -, F - V. P + (',. (3)

n in

(mn mnI' +mnej) + V. (v 2 v I' +m ne})

- env. F- V-(vP) - V.q + Cw. (4)

The first Maxwell equation for the electric potential must be adjoined; each species con-

tributes a corresponding moment subsystem, with appropriately signed charge. We begin the

derivation with the definitions and assumptions. The concentration is given by n := f f du;

the average velocity by v := -1 f uf du; the momentum by p := mnv; the random velocity

by c := u - v; the pressure tensor by Pj := m f cicjf du; and the internal energy density by

el := _ f I c I2 f du. This function represents energy/unit mass/unit concentration. The

heat flux q is given by qi : - f c, I c 12 f du. Finally, for reference in subsequent subsec-

tions, the electron current density is given by J := -env, and the energy flux is given by

S := fu{ I u 21/ du. The assumptions on f are now stated. The function f is assumed

to decrease sufficiently rapidly at infinity:

lim hi(u)f(u) = 0, i = 0, 1,2.

The derivation of (2), (3), (4) proceeds by multiplying the Boltzmann equation (1) by

h0 , hi, and h2, respectively, and integrating over group velocity space. With the application

of certain standard identities ([16]), the mass/momentum/energy system is obtained. In

addition to these transport equations, we have Poisson's equation for the electric field, whcrc

nd := doping and c := dielectric:

F = -Ve, (5)

V.(cVO) = -Ecn,- (6)

Here, we have used the convention that there are different species, each ,,i concentration ni

and charge e2 . The entire system consists of equations (2), (3), (4) repeated according to

species, and (5), (6).

2.2 Moment closure and relaxation relations

The system derived in the preceding subsection has fif'een dependent variables in the case

of one species, determined by q, n, v, P, el, and q. By moment closure is meant the

selection of compatible relations among these va.iables, so that the number of equations is

equal in number to the remaining primitive variables selected. The relations to follow are

characterized by the isotropic/parabolic energy band assumption. We begin by introducing

a new tensor variable T, the carrier teiaperature, defined by

lPi = nkTi

where k is Boltzmann's constai!t, and a scalar variable W, the total carrier energy. A

program of reduction to a set of basic variables, n, v, W, and €, or a set equivalent to these,

can be implemented by the following assumptions:

e The pressure tensor is isotropic, with diagonal entries P, and off-diagonal entries zero,

for a suitable scalar function, P,. P, is related to el via mne, = P.

* It follows from the previous assumption that temperature may be represented by a

scalar quantity T, and that the internal energy is represented in terms of T by

me, = 3kT.2

9 The total energy density (per unit concentration) w is given by combining internal

energy and parabolic energy bands with m assumed constant:

1w = me, + _m I v 12,

2

and the total energy (per unit volume) W is the product, W = nw.

* The heat flux is obtained by a differential expression involving the temperature:

q = -KVT.

Here, K is the thermal conductivity governed by the Wiedemann-Franz law (cf. [2]),

described byS + = 2+ T ( T)r. (7)

2 e T

The standard choice for r is r = -1, but this has some associated difficulties. This

will be amplified later in the paper. Here we simply remark that the term raised to

the exponent r in (7) is proportional to the mobility, which in turn is proportional to

the monmenturn relaxation time.

In the case of N species, the closure relations determine (d + 2)N + 1 variables in d

spatial dimensions. It is possible to rewrite the system (2, 3, 4) with the closure assumptions

incorporated. \Ve have the following.

On (8)

O- + c(V.p) + (p. V)v = -cnF - V(nkT) + Cp, (9)

-- + V. (z ,W) = -cn. F - V. (-vnkT)

+V- (KVT) + Cw'. (10)

The final step deals with the replacement of the collision moments. Motivated by theapproach of [18], [1], [20], and [11], we define the recombination rate R and the momentum

and energy relaxation times, rp and -r, respectively, in terms of averaged collision moments

as follows.

1. The particle recombination rate R is given by

R := -C J := -fC du.

2. The momentum relaxation time r- is given via

:= - m uC du := -Cp.Tp J

3. The energy relaxation time , is given via

W W := I u 2f du:= Cw.

Here, Wo denotes the rest energy, 3kTo, where To is the lattice temperature.

The forms for the relaxation times used in [1] and retained by subsequent authors are:

P = CP(T), (ii)T 1

,,w = CTT + r (12)

Here, cp and c,, are physical constants, and the standard choice for r, just as in (2.7), is -1.

2.3 Drift-diffusion model

The drift-diffusion model may be obtained by taking zeroth order moments of the BTE

and adjoining the Poisson equation. Thus, one obtains the system for N carriers with

concentrations ni and charge ei, I = 1,..., N:

011 + V-J, =-R,, (13)Ot

F =-VO, (14)

V. (cV6) = - E cin, - nd. (15)

There still remains the issiie of determining the constitutive current relations. Classical

drift-diffusion theory gives, for N = 2, n, = i, and n2 =

J -,, nV + cDnVn, (16)

Jp = -tppV6- eDpVp. (17)

5

The introduction of exponential relations for n and p is also common, as is the use of the

Einstein relations linking the mobilities, p,,, pp, and the diffusion coefficients D,, Dp. These

relations are specified by

D.= (Te)p., (18)

D,= (kT/e)yp. (19)

It is also possible to derive the constitutive relations (16), (17), from the first order moment

relations under the assumption that the momentum relaxation times tend to zero. The

details are given in [20]. In fact, the constitutive relations include a heat flux term as well,

which is suppressed at constant temperature. If it is not suppressed, one has an energy drift-

diffusion model. In this derivation, one uses the definition of mobility in terms of relaxation

time.

3 RT Models

In this section, we shall employ a microscopic assumption upon the momentum relaxation

time, viz. , we shall assume that the collision term C in (1) is of the form,

C = -Tp, (20)

where fi is the odd part of f. Note that this contrasts with the macroscopic assumption onr., employed in the hydrodynamic model as described in Section 2. There, the representation

defining -rp was a post averaged expression. Here, the expression is employed in the averaging.

In this case, we may obtain an expression for the energy flux S:

S = -[nPEF + V(nDE)], (21)

where pE and DE are tensor expressions for mobility and diffusion, defined in terms of

moments, and E represents average energy per unit concentration. The details are furnished

in [5]. It is also shown there that the current density has the usual drift-diffusion form, withtensor expressions for mobility and diffusion. The RT model makes the following microscopic

assumptions, with distinction between E and its average, E.

1. The even part of f is isotropic, and a function of . alone, and the relaxation time is

an inverse power function of C:

fo MEo(),

7, r(,) = CE.

6

2. The microscopic kinetic energy is a quadratic function of $, and the mass is not assumed

constant:G() := - I U 12 = E + CE2 , (22)

2where a is an appropriate fitting parameter.

3. The temperature is a modified variable in terms of which the following constitutive

relation holds for E:35

E = 3kT(1 + -akT). (23)22

Equation (23) allows for nonparabolic energy bands as well as non-Maxwellian distributions.

Altogether, the model may be written in terms of the Poisson equation, (6), in conjunction

with the system,

V. J = 0, (24)

V.S = J- F - n (- u). (25)at

Here, J and S have been described previously, the latter in (21). In the expressions for J

and S, the assumptions made for the model lead to scalar representations for the mobility

and diffusion coefficients. For example, the choice made in [5] leads to

p = oTo/T, (26)

= (1- kT)pkT. (27)

The diffusion coefficients are defined by Einstein's relations. The collision term in (25) is

a specified quadratic function of T. One significant advantage of the microscopic relax-

ation time (RT) assumption is that certain key parameters may be fitted via Monte-Carlo

preprocessing, en .;uring reliability of their values.

4 Mathematical Results for the Hydrodynamic Model

In this section, we shall describe some recent mathematical results, obtained in one spatial

dimension. In the first subsection, we shall present existence and boundary layer results for

a simplified version of the steady state hydrodynamic model. This will be followed by a

convergence analysis for Newton's method in the subsonic case.

4.1 Existence and boundary layer theory

We first write down the one dimensional evolution system in the case of a single carrier, in

the absence of recornbination.

01? 0(ni)- + 0X - 0, (28)

7

Op 0(pv + knT) P- + Ox = _-enF- - (29)at x 7pI

OW O(vW + vknT) = -envF + O((OT)I(Ox)) W- W, (30)Ox Ox r(

aFc--T = -en- nd. (31)

Ox

The corresponding steady state system is obtained by setting the time derivatives equal to

zero.

If we rewrite the second steady state equation by use of the pressure, P, we obtain the

equation,O(pv + P) - enF- P. (32)

Ox lrp

The approach of (81 is to eliminate the energy equation (30) from the system, and replace

its role by a relation in the spirit of gas dynamics, i. e. by the constitutive relation,

P(n) = Kn ", y> 1. (33)

When units are selected in which e = 1, c = 1,m = 1, and K = 1, we obtain the system, in

which n and q are the only dependent variables,

j := fly constant, (34)

(F(n)). :=(L-+n )x - -+ n-) :=-S(¢x,n),, (35)n Tp

€ = n-nd. (36)

One can nominally specify boundary conditions on n and 0 at the endpoints of the device,

taken here as the interval, [0, 1]. If the doping is such that the built-in potential is the same

at both ends of the device, then we may take,

0(0) 0, 041) = 01, (37)

for € andn(0) = no, n(1) = ni, (38)

for n, where 44 must satisfy the following consistency relation with respect to j, no, and nj:

f(niJ) - f(no,J) [1 dx (39)

<Jo =n(n,j)n(x)"

Here,j2

f(nj) - 2 + In -

S

It is shown in [8] that a weak solution exists for the system (34), (35), (36), satisfying

the boundary conditions exactly, or, in lieu of this, satisfying precise limiting relationships.

The result for 0 is classical, because the equation is elliptic. The result for n is provisional,

and is detailed now. If J > 0, there is a weak solution n such that the relation,

n - G + a, (41)

holds, where G is H61der (1) continuous, and a is monotone increasing. Although n need

not be of bounded variation, it is an entropy solution, in the sense that the function of x,

H(n(x)) = (F(n(x)) - F(ni,))signum(n(x) - lin,,) + Cx, (42)

is monotone increasing. Here, nmi, is a minimum (location) for F, and C = sup S over

relevant arguments. The following precise statement is available for subsonic boundary

conditions. If no, i, > nmin, then the following holds.

* Either

n(1) =nl,

* or

lim n(x)

exists, and is a supersonic value, i. e. , is less than nmin, and even less than the conjugate

value of n1.

e Similarly, either

n()= no,

* or

lim n(x)

exists and is not less than nin.

In the second instance of both cases above, boundary layers occur, the one on the right

involving transition through the supersonic regime.

4.2 Newton convergence theory

In this subsection, we order the basic variables as v, n, T, and 0, because of symmetry

considerations. Dirichlet boundary conditions are imposed, on n, T, 0, with n(xin) =

n(xmax). In one dimension, the steady state equations are obtained from (28), (29), (30), arid

(31) by setting the time derivatives equal to zero. Dirichlet boundary conditions are imposed

9

in this subsection on it, T', and 0, with fl(0Crijn) ==i(Xrnai). Since, by the conservation of

mass equation, nu j it follows that the boundary conditions for v are periodic. The map

defined by bringing all terms to the left hand sidle of the steady state system is called 0. The

linearized equations thus assume the form, whiere the boundary conditions are homogeneous

Dirichiet conditions for 6n, 61I', and 60, and the boundary conditions on 6v are prescribed

to be periodic,

F 0 [6V 16V]0x r~[ , 1B n + F F 65n (43)

The (spatially) dlependent ei~envalues of the synmictric matrix A are calculated to be

(T 2 )ii 2Ai +in 4r (44)

A1 4145)and the Illaller eigeiivalue is positive if it and T' are strictly positive, and if

2 2V < C(46)

rlbs, typ" of 1)(OIiit in1 ftuict(in )pace is termnel a subsonic p'.tint. This case was first considered

in [91, where dlaiiped Newton/standard finite difference m-ethods were presented. W'hen

Newton's method is employed in this way, it is essential to determine conditions under

which the linlearl incremnents are appropriatex, bou~nded. This is equivalent to uniform bounds

for the operator derivattive inverse maps, and represents one of the three properties for an

(exact) operator Newton miethod to yield existence of a root and ft-quadratic convergence.

The other two are sufficient regularity, anid a sufficiently small starting residual, as measured

in the range space norm. Explicit representations of B, C, D and of F, G, H are given in

[141]. Moreover, if thle System mnap (D, sub~ject t( ap~propriate Dirichlet boundary conditions

onl II, T', anid 0. anid periodic boundary conditions onl v, accordingly 1las the domain D'j C

X --Hill' xM IV~ bE2 'an range inI V 1 L", then (46) will hold for every element

inl it Closed hil 13r0 c X. (-enteredl at a subsonic element u0 E X , such that it and T' are

St rictly p~ositive(, if r0t Is, Sufficiently simall. It Is aippropriate to ass'irne at the outset, then,

'r hat AP1 C- B,~. so t i at every function point in D~ satisfies (46), we mna also assume that

niiiand TI aire iiformly hotunded away f-omi zero) it) this set.

Tlie i.TpScliit / propertyv of tll'Im an V', whe(re here we view the system (.13) as the repre-

senltaion) for (1%' 11. 7'T,( .) f with

b'm'. ~ O j.( ), f = (fl 12h), (47)

is evident from the representation for I'.

The uniform inverse bounding proceeds as follows. As shown in [9], with the function

spaces selected in this paper, the H' product norm of z can be estimated in terms of the

L2 norms of L,,, w', and f, under the conjunction of the hypothesis (46) and the L2 x L'

coerciveness assumption,

A = E + E* - d is uniformly positive definite. (48)

dx

Here, E' is the matrix transpose of E and the latter is defined by

r dn dvE- dv 1 {-1 n (49)

Tx + 1p i n dxI

A final calculation, making use of the inner product of [0,w] with (43), and the hypothesis,

h1l is positive, and sufficiently large, (50)

where hi1 is the nonzero entry of H, given by

+ dT - (W - Wo)-,w (51)dx 7-.2

with -' =-1 shows that the I product norm of [z, w] is estimated in terms of the L norm- dT s

of f. This series of calculations controls the L' norm of [z,co]; the L- norm of [z',J"] is

now estimated by direct use of the system (43), making use of the fact that [v, nT,' ] E B,0 .

We have now outlined the proof of the following.

Theorem 4.1 Let the function spaces X and Y be selected as above, let the steady state

system map (D be given with Dirichlet boundary conditions on n, T, 0, and periodic boundary

conditions on v, such that Dp C B,0, where every point in B o is a subsonic point, with uni-

form positivity bounds. If (48) and (51) hold, and xO is such that 4(Xo) is sufficiently small,

then an R-quadratically convergent Newton sequcnce {x,} may be defined in the standard

way, with limit x, satisfying (,(x) = 0.

11

5 Discrete Schemes Based on Adaptive Stencils: ENO

In this section, we shall briefly describe the ENO schemes as developed in [24] and [25].

Consider a system of hyperbolic conservation laws of the form

d

Ut + -if(u), g(u,x,t), (52)1

whereU =- (Ul,'',Um)T ,

X 7, (Xl,'''Xm),

and the hyperbolicity condition,

- is diagonalizable, with real eigenvalues,1 (O

holds for any real -= ( -1," , d). An initial condition is adjoined to (52).

For systems of conservation laws, local field by field decomposition is used, to resolve

waves in different characteristic directions. Analytical expressions are employed for the

eigenvalues and eigenvectors of an averaged Jacobian matrix. Typically, the Roe average

[19] is employed. One feature of the ENO schemes in [24] and [25], which is distinct from

the original ENO schemes of Harten et al [12], is that multidimensional regions are treated

dimension by dimension: when computing fi(u), in any particular direction, variables in all

other directions are kept constant, and the Jacobians are treated in this direction. This, in

essence, reduces the determination of the scheme to the case of a single conservation law in

one spatial dimension. Thus, to describe the schemes, consider the scalar one dimensional

problem, and a conservative approximation of the spatial operator given by1

L(=)j --I( fj+ m - ). (53)

Here, the numerical flux f is assumed consistent:

+= Uj-,,... ,uj+k); f(u,...,u) = f(u). (54)

The conservative scheme (53), which characterizes the f divided difference as an approx-

irnation to f(u)1 , suggests that f can be identified with an appropriate function h satisfying

f( _- - h( ) d . (55)2

If 11 is any p1ri1nitive of h, then h can be computed from H'. H itself can be constructed

by Nv,,wton' s divide.d differeice method, beginning with differences of order one, since the

12

constant term is arbitrary. The necessary divided differences of H, of a given order, are

expressed as constant multiples of those of f of order one lower. After the polynomial Q of

degree r + 1 has been constructed, set

]j+ = d Q(x)i= , (56)

to obtain an rth order method. The construction is based on an adaptive stencil in the

following sense:

9 One begins with an appropriate starting point to the left or right of the current "cell"

by means of upwinding as determined by the sign of the derivative of a selected flux.

e As the order of the divided differences is increased, the divided differences themselves

determine the stencil: the "smaller" divided difference is chosen from two possible

choices at each stage, ensuring a smoothest fit.

* Lax-Friedrichs building blocks or Roe building blocks can both be used. For the lat-

ter, in cells with sonic points, a local Lax-Friedrichs building block is used to avoid

expansion shocks.

Steady states are reached by explicit time stepping of arbitrary order; nonstandard high

order Runge-Kutta methods exist [24] which preserve nonlinear stability of the first order

Euler forward version under suitable CFL time step restrictions. The computer program

is fully vectorized for computations on Cray supercomputers. For details of the efficient

implementation, see [23].

6 Conservation Law Format for Hydrodynamic andRT Models

In this section, we shall specify the conservation law format for the two dimensional hydro-

dynamic model, and for the one dimensional RT model.

6.1 Hydrodynamic model conservation format

Define the vector of dependent variables as

U = (n,, , W), (57)

where p = (a,r). The system (8), (9), (10) can be written in the concise form, in two

dimensions, as

ut + f, (u), + f 2(u)v = c(u) + G(u, ¢) + (0,0,0, V. (KVT)). (58)

13

The following identifications have been made in (58).

o, 2 a 2 - 2 a7 aW o2 +T

2

fi(U) = (m a, 3 i + W ), n' 3mn 3m 2 n 2 (59)

,T ar 2 7 2 ar2 5-rW or2 +7 2m m 3 mn 2ra n r 3ran a3nn-----

f-, =(n rn (-±m + W- 2man ) ' 3mn -T 3rn2n2 ),(60)

c(u) = (0, a T W-Wo) (61)

Tp 7p Tw

G(u) = (0,-cnF1,-enF2, -enF. v). (62)

The eigenvalues and eigenvectors of f' and f2 are known (cf. [23]), and are readily

incorporated into the field by field decomposition required for the implementation of ENO.

6.2 RT conservation format

We shall present the conservation law form of the RT model. We begin with the vector form,

ut + f(u). = g(u).. + h(u). (63)

In equation (63),

U = (en, nE (64)m

f(u) = O'n (ey(E), ,iE(E) + D(E)), (65)

g(u) = (nD(E), nDE(E)), (66)

h(u) = (0, enid(E)(') 2 + e(n - nd)nD(E) - n((67)

It can be shown that the left hand side defines a hyperbolic system, since the eigenvalues of

f'(u) are real, for all positive n and T.

7 Numerical Simulation Results

We now present numerical simulation results for one carrier, two dimensional MESFET

devices and one dimensional diodes. The third order ENO shock-capturing algorithm with

Lax-Friedrichs building blocks, as described briefly in Section 5 and in more detail in [25], is

applied to the hyperbolic part (the left hand side) of Equations (6.2) and (6.7). A nonlinearly

stable third order Runge-Kutta tinw, discretization [24] is used for the time evolution towards

steady states. The forcing terms on the right hand side of (6.2) and (6.7) are treated in a

time consistent way in the Runge-Kutta time stepping. The double derivative terms on the

14t

right hand side of (6.2) and (6.7) are approximated by standard central differences owing to

their dissipative nature. The Poisson equation (2.6) is solved by direct Gauss elimination for

one spatial dimension and by Successive Over-Relaxation (SOR) or the Conjugate Gradient

(CG) method for two spatial dimensions. Initial conditions are chosen as n = nd for theconcentration, T = To for the temperature, and u = v = 0 (two spatial dimensions) or u = 0

(one spatial dimension) for the velocities. A continuation method is used to reach the steady

state: the voltage bias is taken initially as zero and is gradually increased to the required

value, with the steady state solution of a lower biased case used as the initial condition for

a higher one.

7.1 Two dimensional MESFET

We simulate, using the Hydrodynamic model (6.2)-(2.6), a two dimensional MESFET of the

size 0.6 x 0.2rn2 . The source and the drain each occupies 0.1,rm at the upper left and the

upper right, respectively, with the gate occupying 0.2jim at the upper middle (Figure 1, left).

The doping is defined by nd = 3 x 1017cm- in [0, 0.1] x [0.15, 0.2] and in [0.5, 0.6] x [0.15, 0.2],

and 7d = I x 101 7 cm - ' elsewhere, with abrupt junctions (Figure 1, right). A uniform grid

of 96 x 32 points is used. Notice that even if we may not have shocks in the solution, the

initial condition n = 7d is discontinuous, and the final steady state solution has a sharp

transition around the junction. With the relatively coarse grid we use, the non-oscillatory

shock capturing feature of the ENO algorithm is essential for the stability of the numerical

procedure.

surce .~ d',,

/[.

4

Figure 1: Two dimensional MESFET. Left: the geometry; Right: the doping lid

We apply, at the source and drain, a voltage bias vbias = 2V. The gate is a Schottky

1,5

contact, with a negative voltage bias ugat = -O.81' and a verv low coiwceltration value

n = 3.9 x 105cm - 3 obtained from Equation (5.1-19) of 122]. Thc lattice tzniperaturc

taken as To = 300'K. The numerical boundary conditions are summarized as follows (where

= k-T In (a) with kb = 0.138 x 10- 4 , r = 0.1602, and n= 1.4 x 101°CrII - 3 in our units):

* At the source (0 < x < 0.1,y = 0.2): (D = (Do for the potential; n = 3 x 1017crn-

for the concentration; T = 300°K for the temperature; u =- Otmn/ps for the horizontal

velocity; and Neumann boundary condition for the vertical velocity v (i.e. 2-- = 0

where it is the normal direction of the boundary).

* At the drain (0.5 < x < 0.6, y = 0.2): D = 40 + vbias = (Do + 2 for the potential;

n = 3 x 1017c - 3 for the concentration; T = 300'K for the temperature; u = Om/ps

for the horizontal velocity; and Neumann boundary condition for the vertical velocity

V.

* At the gate (0.2 < x < 0.4, y 0.2): = 4)o + vgate = (Do - 0.8 for the potential;

n = 3.9 x 10 5crn- 3 for the concentration; T = 300'K for the temperature; u = Oprn/ps

for the horizontal velocity; and Neumann boundary condition for the vertical velocity

V.

a At all other parts of 'the boundary (0.1 ' x < 0.2,y = 0.2; 0.4 < x < 0.5,y = 0.2;

x = 0,0 < y < 0.2; a = 0.6,0 < y - 0.2; and 0 < x < 0.6,y = 0), all variables are

equipped with Newmann boundary conditions.

The boundary conditions chosen are based upon physical and aumerical considerations.

They may not be adequate mathematically, as is evident from some serious boundary layers

observable in Figures 2 through 6. ENO methods, owing to their upwind nature, are robust

to different boundary conditions (including over-specified boundary conditions) and do not

exhibit nunierical difficulties in the presence of such boundary layers, even with the extremely

low concentration prescribed at the gate (around 1012 relative to the high doping). We point

out, however, that boundary conditions affect the global solution significantly. We have also

simulated the same problem with different boundary conditions, for example with Dirichlet

boundary conditions everywhere for the temperature, or with Neumann boundary conditions

for all variables except for the potential at the contacts. The numerical results (not shown

in this paper) are noticeably different. This indicates the importance of studying adequate

boundary conditions, fLon both a physical and a mathematical point of view.

In Figures 2 through 6, we show pictures of the concentration n, temperature T, horizontal

velocity u, vertical velocity v, and the potential 4). Surfaces of the solution are shown at

16

the left, and cuts at y = 0.175, which cut through the middle of the high doping "blobs"horizontally, are shown at the right.

N

30000 '

20000(

100000

00 0.2 0.4 0.6

Figure 2: Two dimensional MESFET, concentration n. Left: surface of the solution;Right: cut at y 0.175

T

T

200C

1000c

0 . . . [I _.0 0.2 O. 0.6 -

Figure 3: Two dimensional MESFET, temperature '. Left: surface of the solution;

Right: cut at y = 0.175

17

UU

0.10

0.05

0.00

"0.10 r"F

0.2 0.4 0.6$

Figure 4: Two dimensional MESFET, horizontal velocity u. Left: surface of the solution;

Right: cut at y = 0.175

V

0.2 ~

0.

0.0

"0,2

-01

020 0.2 0.4 0. ,

Figure 5: Two dimensional MESFET, vertical velocity v. Left: surface of the solution;

Right: cut at y = 0.175

18

PHi

~/

* .-

/

0.0 i-

0 0.2 0.4

Figure 6: Two dimensional MESFET, potential (. Left: surface of the solution; Right:

cut at y =0.175

Notice that there is a boundary layer for the concentration n at the drain but not at the

source. Also notice the rapid drop of n at the depletion region near the gate. The temperature

achieves its maximum around the left corner of the drain. The leakage current at the gate

appears negligible from the normal velocity component, while the horizontal component

shows evidence of strong carrier movement toward the source beneath the left gate area, and

strong movement toward the drain immediately to the left of the drain junction.

We have also simulated the same MESFET with a higher doping ratio: 3 x 1017 cm- 3 in

the high doping region versus 1 x 10cIm- in the low doping region. We observe similar

results (pictures not shown here).

7.2 HD model for a one dimensional diode - spurious velocityovershoot

A notorious plhenomenorn of lID models is that spurious velocity overshoot occurs at the

drain junction of an 1+-1-1 + diode. It is intrinsic to tlie model and is not a numerical

artifact, as is verified by our grid refinement study and 'y using different numerical algo-

rithms. This phenonenon is closely related to the physical assumption governing the heatconduction term. (',iini, Odeh and Rudan [10] observed that the spurious overshoot can be

greatly reduced by an empirical modification of the Wiedemann-Franz law for the thermal

conductivity.

In this subsection we perform an extensive numerical study of the dependency of the

19

spurious velocity overshoot upon the heat conduction term. The n+-n-n+ diode we simulate

has a length 0.61an, %,with a doping defined by Id = 3x 107 cnz - in [0, 0.1] and in [0.5,0.6], and

71 = 1 x 10 'cr - 3 in [0.15, 0451, with smooth junctions (Figure 7). The lattice temperature

is taken as 1o 296.21K. We apply a voltage vbias 1.5V, as is the case in [10].

0 C, C, ;, (- -. . .

r /

II

F'ire 7: ''i 11 4 iig 1 d for the one dimiensional n+-n-n+ diode

90

0.20

0.10 F-0.05

0 0.2 0, 04

",,r

..-. 0000 -

7;.--0--CO

000.2 0.4 0.6 0.00-

25 .2

4000 --- 30 0 0-

300000

)cooF

CCL

y__ .. I -1 - I £

0.2 0.4 0.6 0.2 0.4 0.b

Figure 8: IID for one dimensional n+-rin+ diode. Velocity u (upper), temperature T

(middle left), concentration n (middle right), total energy W (lower left) and electric field

-0' (lower right). Solid line: r=-1; dashed line: r=~-2; circles: r=-2 in the coefficient of (2.7);pluses: r=-2 in (2.7)

21

The standard LID model uses r = -1 in the Wiedemann-Franz law (2.7) and the relax-

ation times (2.11). The numerical solution of this is shown as solid lines in Figure 8. We

can clearly observe the spurious velocity overshoot at the right junction, but otherwise the

solution is basically correct comparing with direct Monto-Carlo simulations (not shown).

When r is taken as -2 in (2.7) and (2.11), the solution is completely wrong (dashed line in

Figures 8). However, when one takes r = -2 only in the coefficient of K in (2.7) but leaves

r = -1 in the power of sc in (2.7) and in (2.11), i.e., when one uses

1 k2 PO T (T (68)

in the place of (2.7) and leaves r = -1 in (2.11) unchanged, as was done in [10], one obtains a

greatly reduced spurious overshoot (the circles in Figures 8). Finally, the result with r = -2

in K in (2.7) but with r = -1 in (2.11) unchanged, is shown by pluses in Figure 8. We can

see that the spurious overshoot also disappears.

7.3 RT model for a one dimensional diode

We present nulerical simulation results for the RT model, described in Section 3, for the

same one dimensional diode used in Subsection 7.2. Although the RT model is a parabolic

system with two equations, the existence of sharp transition regions near the junctions

justifies the usage of ENO dtock capturing algorithms for the hyperbolic part.

In Figure 9, we show the results of velocity u, temperature T, concentration n, total

energy E, and (,letIric iheld -' of the RT simulation, in circles, in a background of standard

LID results (r = - 1 ) in solid lines, and of HID results with r = -2 in the coefficient of K in

(2.7) iut wil Ii r Iv In t lie pow, er (i.e., (2.7) is replaced by (7.1)), and r = -1 in (2.11), in

daslwJ line. \V, ca, see hat the liT model greatly reduces the spurious velocity overshoot

and is coniiwrdhlc with the result of the empirically modified HD result in dashed lines.

Extensive numerical tests about the Rit model, as well as comparisons between the RT

and 1I) models. (,.!stitute ongoing research, jointly with U. Ravaioli, E. Kan and D. Chen

a t '! , , -: :.I I ' , " i i , , is .

Acknowledgements: We would like to thank Edwin Kan, Umberto Ravaioli and Stan-

ley Osher for helpful discussions. Computations were performed on the Cray YMP at the

Pittsburgh Superconlputing (-'enter and on the Cray YMP at the University of Illinois.

22

0.20

0.15

0.10

0.05

0.0cL 70 0.2 0.4 0.6 -

30001

250 50000 "

1500L300.

10VI

2000 1 0000C

50000 ,100000

0'

5000 .

4010.000

-2.5

3000 -.

-5.02000-

-7.51000

- -10.0

X0 0.2 0.4 0. 6

Figure 9: One dimensional n+-n-n+ diode. Velocity u (upper), temperature T (middleleft), concentration n (middle right), total energy E (lower left), and electric field -4Y (lowerright). Solid line: standard HD with r=-1; circles: RT; dashed lines: HD with r=-2 in thecoefficient of (2.7)

23

References

[1] G. Baccarani and M. It.. Wordemian. An investigation of steady-state velocity overshoot

effects in Si and GaAs dlevices. Solid State Elect r., 28:407-416, 1985.

[2] 1-. J. Blatt. Physics of Electric Conduction in Solids. McGraw Hill, New York, 1968.

[3] K. Blotek iaer. Transport equations for electrons in two-valley semiconductors. IEEE

Trans. Electron Devices, 17:38-47, 1970.

[4] C.. Cercignani. The iBoltzinann Equation and its Application. Springer- Verlag, New

York, 198 7.

[51 D. Chen, E. Kan, K. Hess, and U. Ravaioli. Steady-state macroscopic transport equa-

tions and coefficients for submicron device modeling. To appear.

[6] E. Fatemii, C. Gardner, J. Jerome, S. Osher, and D. Rose. Simulation of a steady-state

electron shock wave in a submicron semiconductor device using high order upwind meth-

ods. In 1K. Hess, J. P. Leburton, and U. Ravaioli, editors, Computational Electronics,

pages 27-32. IKluwer Academ-ic Publishers, 1991.

[7] E. Faternil. .J. Jerome, and S. Osher. Solution of the hydrodynamic device model using

high-order nonoscillatory shock capturing algorithms. IEEE Transactions on Computer-

Aidtd Dtsiqn of Integraitd CI rcuits and Systems, CAD-10:232-244, 1991.

[8] Irene NM. Ganiba. Stationary transonic solutions for a one-dimensional hydrodynamic

nmodel f'or semiiconiductors. Commnunications in P. D. F.

[9] C-. L Guadnr, J. \V. Jerome, and D. J. Rose. Numerical methods for the hydrody-

naiilc device mo1del: Subsonic flow. IEEE Transactions on Computer-Aided Design of

Int(yrat(d( Circtits alnd Systemns. CAD-S:501-507. 1989.

1,011A Giludli. 1'. Odch, and 'I. lRudan. An efficient, discrctization scheme for the energy

cont in iii v equation1 in semiconductors. In Proceedings of SISDP, pages 387-390, 1988.

[11] \V. lfiisch and N1. %liura-NMatta:, (ch. The hot-electron problem in small semiconductor

dle v ic cs. ppl. Phys., 60:6,50-656, 1986.

[121 A\. I lartenit. I'I ngquist. S. Osher and S. Ch'lakravarthy. Uniformly high order accurate

esseilt iallY non-osci Ilat ory schemecs, Ill. J. Comnp. Ph ys., 71:231-303, 1987.

2-1

[13] Joseph W. Jerome. Consistency of semiconductor modelling: An existence/stability

analysis for the stationary van Roosbroeck system. SIAM J. Appl. Math., 45(4):565-

590, August 1985.

[14] Joseph W. Jerome. Algorithmic aspects of the hydrodynamic and drift-diffusion mod-

els. In R. E. Bank, R. Bulirsch, and K. Merten, editor, Mathematical Modelling and

Simulation of Electrical Circuits and Semiconductor Devices, pages 217-236. Birkhauser

Verlag, 1990.

[15] Joseph W. Jerome and Thomas Kerkhoven. A finite element approximation theory for

the drift-diffusion semiconductor model. SIAM J. Num. Anal., 28:403-422, 1991.

[16] Joseph W. Jerome and Thomas Kerkhoven. Steady State Drift Diffusion Semiconductor

Models. SIAM, 1992.

[17] M. S. Mock. On equations describing steady-state carrier distributions in a semicon-

ductor device. Comm. Purc Appl. Math., 25:781-792, 1972.

[18] J. P. Nougier. J. Vaissiere, D. Gasquet, J. Zimmermann, and E. Constant. Determina-

tion of the transient regime in semiconductor devices using relaxation time approxima-

tions J. Appl. Phys., 52:825-832, 1981.

[19] P. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J.

Comp. Phys., 27:1-31, 1978.

[20] M. Rudan and F. Odeh. Multi-dimensional discretization scheme for the hydrodynamic

model of semiconductor devices. COMPEL, 5:149-183, 1986.

[21] T. Seidman. Steady state solutions of diffusion reaction systems with electrostatic

convection. Nonlinear Anal., 4:623-637, 1980.

[22] S. Selberherr. Analysis and Simulation of Semiconductor Devices. Springer-Verlag,

Wien - New York, 1984.

[23] C.-W. Shu, G. Erlebacher, T. Zang, D. Whitaker, and S. Osher. High-order ENO

schemes applied to two- and three-dimensional compressible flow. ICASE Report 91-38,

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[24] C.-W. Shu and S. J. Oslier. Efficient implementation of essentially non-oscillatory shock

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25

[25] C.-W. Shu and S. J. Osher. Efficient implementation of essentially non-oscillatory shock

capturing schemes, II. J. Comp. Phys., 83:32-78, 1989.

26

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ENERGY MODELS FOR ONE-CARRIER TRANSPORT IN SEMICONDUCTOR

DEVICES NASI-18605

6. AUTHOR(S) 505-90-52-01

Joseph W. Jerome

Chi-Wang Shu

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13. ABSTRACT (AMaxmurn 200 words)

Simulations of the hydrodynamic model for semi-conductor device in one and twospace dimensions, and simulations of a newly developed energy model, the RT model,are presented using essentially non-oscillatory (ENO) shock capturing algorithms.Comparisons between models are performed. Some mathematical results regarding thcsemodels are also presented.

14 SUBJECT TERMS 15. NUMBER OF PAGES

enrr~y models; shock capturing algorithms; conservation laws; 28

velocity overshoot; parabolic and nonparabolic energy bands OPrte

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